Dividing Integers

Dividing Integers

What You Will Learn

-Analyze patterns of quotients of integers with same and different signs, recognizing division by zero as undefined.

-Divide integers.

-Evaluate variable expressions involving integer division.

I. Analyze Patterns of Quotients of Integers with Same and Different Signs, Recognizing Division by Zero as Undefined

Another important step in learning how to compute with integers is learning how to divide them. You can look for patterns in a sequence of quotients just as you looked for patterns in a sequence of products in an earlier lesson. These patterns will help you to understand the rules for dividing integers.

Let’s look at some integer patterns with division. We are looking at quotients. 

A quotient is the answer in a division problem.

Multiple Choice

A quotient is the answer in a _____________ problem.

• addition

• subtraction

• multiplication

• division

Example

Use a pattern to find the missing quotients below.

• 6÷2 = 3

• 4÷2 = 2

• 2÷2 = 1

• 0÷2 = 0

• −2÷2 = ?

• −4÷2 = ?

Open Ended

What is the missing quotient below.

−2÷2 = ?

Open Ended

What is the missing quotient below.

−4÷2 = ?

Look for a pattern among the quotients. 

• Remember that a pattern has a rule that makes it repeat in a certain way. Look at the pattern to the right.

• This pattern uses shapes and not numbers, but there is still a rule that applies, that makes the pattern repeat itself in the way that it does.

Now look at the number pattern.

• You will see that you can subtract 1 from the previous quotient to find the next quotient.

• Rremember, subtracting 1 is the same thing as adding its opposite, -1.

• Try adding -1 to the previous quotients to find the next quotient.

To find the quotient of −2÷2

• divide 2 by 2

• the quotient = 1

• Our rules are the same for division as multiplication.

• We have an ODD number of negative integers ( 1)

• so, our answer will be negative.

• −2÷2 = -1
  

To find the quotient of −4÷2

• divide 4 by 2

• the quotient is 2

• We have an ODD number of negative integers, so our quotient will be NEGATIVE.

• −4÷2 = -2

This shows the completed division facts.

Each quotient is still 1 less than the previous quotient.

What conclusions can we draw from this pattern?

• When a positive integer is divided by a positive integer, 2, the quotient is positive.

• When zero is divided by a positive integer, 2, the quotient is zero.

• When a negative integer is divided by a positive integer, 2, the quotient is negative.

• These are the beginnings of our rules for dividing integers.

Multiple Choice

When a positive integer is divided by a positive integer, the quotient is ____________.

• positive

• negative

• zero

Multiple Choice

When zero is divided by a positive integer, the quotient is ____________.

• positive

• negative

• zero

Multiple Choice

When a negative integer is divided by a positive integer, the quotient is ________________.

• positive

• negative

• zero

Let’s look at another pattern to complete these rules.

• What do you notice about these facts?

Open Ended

What do you notice about these facts?

You may notice the following rules.

• When a positive integer is divided by a negative integer, the quotient is negative.

• When zero is divided by zero, the quotient is undefined, not zero.

• NOTE: Any number divided by zero is considered undefined.

• When a negative integer is divided by a negative integer, the quotient is positive.

Multiple Choice

When zero is divided by zero, the quotient is __________.

• positive

• negative

• zero

• undefined

Based on the patterns, here are the rules for dividing integers.

Notice that the rules for dividing integers are the same as the rules for multiplying integers.

Now, let’s apply these rules to dividing integers.

Example 1

Find the quotient (−33)÷(−3)

• To find this quotient, we need to divide two negative integers.

• Divide the integers without paying attention to their signs.

• 33 ÷ 3 = 11

• The quotient will be positive

• (−33) ÷ (−3) = 11

• The quotient is 11

Example 2

Find the quotient (−20)÷5

• To find this quotient, we need to divide two integers with different signs.

• Divide the integers without paying attention to their signs.

• 20 ÷ 5 = 4

• Give the quotient a negative sign because the signs are different, indicating a negative answer.

• so (−20) ÷ 5 = −4

• The quotient is -4.

These problems used a division sign, but remember we can also show division using a fraction bar where the numerator is divided by the denominator.

Now, it’s time for you to practice applying these rules to figuring out quotients.

Multiple Choice

−12 ÷ −3

• positive

• negative

• zero

• undefined

Multiple Choice

 $\frac{18}{-3}$  

• positive

• negative

• zero

• undefined

Multiple Choice

What movie is this?

• mulan

• Tarzan

• Beauty and the Beast

• Frozen

Multiple Choice

−24 ÷ 0

• positive

• negative

• zero

• undefined

Multiple Choice

The answer in a division problem.

• Quotient

• Product

• Sum

• Difference

Multiple Choice

What song is this?

• Just the way you are

• Your eyes, your eyes

• The way you do

• Shining Stars

Multiple Choice

when an integer is divided by 0, the answer is __________.

• positive

• negative

• zero

• undefined

Multiple Choice

 $\frac{-144}{12}$  

• 11

• -11

• 12

• -12

Multiple Choice

 $\frac{-105}{7}$  

• -14

• -15

• -16

• -17

Multiple Choice

True or False: Michael Jordan was cut from his high school basketball team

• True

• False

Multiple Choice

65 ÷ -13

• 6

• -6

• 5

• -5

Multiple Choice

75 ÷ 15

• 4

• -4

• 5

• -5

Multiple Choice

60 ÷ 12

• 6

• -6

• 5

• -5

Multiple Choice

What is Ms. Peyton's son's name?

• Kolbi

• Mikaela

• Hayes

• Ice

Multiple Choice

-24 ÷ 8

• 3

• -3

Multiple Choice

-21 ÷ 7

• 3

• -3

Multiple Choice

True or False: Applesauce was the first food eaten in space by astronauts?

• True

• False

Multiple Choice

6 ÷ -6

• 1

• -1

Multiple Choice

8 ÷ 4

• 2

• -2

Multiple Choice

-8 ÷ -2

• 4

• -4

Multiple Choice

-24 ÷ 4

• 6

• -6

Multiple Choice

Which color is not a primary color?

• Yellow
• Red
• Purple
• Blue

Multiple Choice

-8 ÷ 4

• 2

• -2

Multiple Choice

35 ÷ -5

• 7

• -7

Multiple Choice

0 ÷ -8

• positive

• negative

• zero

• undefined

Multiple Choice

When dividing integers....

• follow the same rules as multiplying integers
• Follow the same rules as adding integers
• Follow the same rules as subtracting integers
• Always do the opposite

Multiple Choice

To "Go bananas" means _________

• go and buy some bananas

• eat this banana

• become crazy

• become angry

Multiple Choice

If there are an odd number of negatives in a problem, the answer is
Example: (-4)(-5)(-2)

• Positive
• Negative

Multiple Choice

What song is this?

• Hapana

• Hafana

• Havana

• Hanyooo

Multiple Choice

Even number of negatives in a problem means the answer will be

• Positive
• Negative

Multiple Choice

Negative x Negative =

• Positive
• Negative

Multiple Choice

What song is this?

• Delicate

• I like it

• Memories

• Better now

Multiple Choice

Positive x Negative =

• Positive
• Negative

Multiple Choice

Positive x Positive =

• Positive
• Negative

Multiple Choice

What song is this?

• Back to you

• Love lies

• Girls like you

• Don't start now